Question: $\dfrac{ 8u - 6v }{ -9 } = \dfrac{ u + 2w }{ -7 }$ Solve for $u$.
Answer: Multiply both sides by the left denominator. $\dfrac{ 8u - 6v }{ -{9} } = \dfrac{ u + 2w }{ -7 }$ $-{9} \cdot \dfrac{ 8u - 6v }{ -{9} } = -{9} \cdot \dfrac{ u + 2w }{ -7 }$ $8u - 6v = -{9} \cdot \dfrac { u + 2w }{ -7 }$ Multiply both sides by the right denominator. $8u - 6v = -9 \cdot \dfrac{ u + 2w }{ -{7} }$ $-{7} \cdot \left( 8u - 6v \right) = -{7} \cdot -9 \cdot \dfrac{ u + 2w }{ -{7} }$ $-{7} \cdot \left( 8u - 6v \right) = -9 \cdot \left( u + 2w \right)$ Distribute both sides $-{7} \cdot \left( 8u - 6v \right) = -{9} \cdot \left( u + 2w \right)$ $-{56}u + {42}v = -{9}u - {18}w$ Combine $u$ terms on the left. $-{56u} + 42v = -{9u} - 18w$ $-{47u} + 42v = -18w$ Move the $v$ term to the right. $-47u + {42v} = -18w$ $-47u = -18w - {42v}$ Isolate $u$ by dividing both sides by its coefficient. $-{47}u = -18w - 42v$ $u = \dfrac{ -18w - 42v }{ -{47} }$ Swap signs so the denominator isn't negative. $u = \dfrac{ {18}w + {42}v }{ {47} }$